[comment {-*- tcl -*- doctools manpage}]
[manpage_begin math::linearalgebra n 1.1]
[copyright {2004-2008 Arjen Markus <arjenmarkus@users.sourceforge.net>}]
[copyright {2004 Ed Hume <http://www.hume.com/contact.us.htm>}]
[copyright {2008 Michael Buadin <relaxkmike@users.sourceforge.net>}]
[moddesc   {Tcl Math Library}]
[titledesc {Linear Algebra}]
[category  Mathematics]
[require Tcl [opt 8.4]]
[require math::linearalgebra [opt 1.1]]

[description]
[para]
This package offers both low-level procedures and high-level algorithms
to deal with linear algebra problems:

[list_begin itemized]
[item]
robust solution of linear equations or least squares problems
[item]
determining eigenvectors and eigenvalues of symmetric matrices
[item]
various decompositions of general matrices or matrices of a specific
form
[item]
(limited) support for matrices in band storage, a common type of sparse
matrices
[list_end]

It arose as a re-implementation of Hume's LA package and the desire to
offer low-level procedures as found in the well-known BLAS library.
Matrices are implemented as lists of lists rather linear lists with
reserved elements, as in the original LA package, as it was found that
such an implementation is actually faster.

[para]
It is advisable, however, to use the procedures that are offered, such
as [emph setrow] and [emph getrow], rather than rely on this
representation explicitly: that way it is to switch to a possibly even
faster compiled implementation that supports the same API.

[para]
[emph Note:] When using this package in combination with Tk, there may
be a naming conflict, as both this package and Tk define a command
[emph scale]. See the [sectref "NAMING CONFLICT"] section below.

[section "PROCEDURES"]

The package defines the following public procedures (several exist as
specialised procedures, see below):



[para]
[emph "Constructing matrices and vectors"]

[list_begin definitions]

[call [cmd ::math::linearalgebra::mkVector] [arg ndim] [arg value]]

Create a vector with ndim elements, each with the value [emph value].

[list_begin arguments]
[arg_def integer ndim] Dimension of the vector (number of components)
[arg_def double value] Uniform value to be used (default: 0.0)
[list_end]

[para]
[call [cmd ::math::linearalgebra::mkUnitVector] [arg ndim] [arg ndir]]

Create a unit vector in [emph ndim]-dimensional space, along the
[emph ndir]-th direction.

[list_begin arguments]
[arg_def integer ndim] Dimension of the vector (number of components)
[arg_def integer ndir] Direction (0, ..., ndim-1)
[list_end]

[para]
[call [cmd ::math::linearalgebra::mkMatrix] [arg nrows] [arg ncols] [arg value]]

Create a matrix with [emph nrows] rows and [emph ncols] columns. All
elements have the value [emph value].

[list_begin arguments]
[arg_def integer nrows] Number of rows
[arg_def integer ncols] Number of columns
[arg_def double value] Uniform value to be used (default: 0.0)
[list_end]

[para]
[call [cmd ::math::linearalgebra::getrow] [arg matrix] [arg row] [opt imin] [opt imax]]

Returns a single row of a matrix as a list

[list_begin arguments]
[arg_def list matrix] Matrix in question
[arg_def integer row] Index of the row to return
[arg_def integer imin] Minimum index of the column (default: 0)
[arg_def integer imax] Maximum index of the column (default: ncols-1)

[list_end]

[para]
[call [cmd ::math::linearalgebra::setrow] [arg matrix] [arg row] [arg newvalues] [opt imin] [opt imax]]

Set a single row of a matrix to new values (this list must have the same
number of elements as the number of [emph columns] in the matrix)

[list_begin arguments]
[arg_def list matrix] [emph name] of the matrix in question
[arg_def integer row] Index of the row to update
[arg_def list newvalues] List of new values for the row
[arg_def integer imin] Minimum index of the column (default: 0)
[arg_def integer imax] Maximum index of the column (default: ncols-1)
[list_end]

[para]
[call [cmd ::math::linearalgebra::getcol] [arg matrix] [arg col] [opt imin] [opt imax]]

Returns a single column of a matrix as a list

[list_begin arguments]
[arg_def list matrix] Matrix in question
[arg_def integer col] Index of the column to return
[arg_def integer imin] Minimum index of the row (default: 0)
[arg_def integer imax] Maximum index of the row (default: nrows-1)
[list_end]

[para]
[call [cmd ::math::linearalgebra::setcol] [arg matrix] [arg col] [arg newvalues] [opt imin] [opt imax]]

Set a single column of a matrix to new values (this list must have
the same number of elements as the number of [emph rows] in the matrix)

[list_begin arguments]
[arg_def list matrix] [emph name] of the matrix in question
[arg_def integer col] Index of the column to update
[arg_def list newvalues] List of new values for the column
[arg_def integer imin] Minimum index of the row (default: 0)
[arg_def integer imax] Maximum index of the row (default: nrows-1)
[list_end]

[para]
[call [cmd ::math::linearalgebra::getelem] [arg matrix] [arg row] [arg col]]

Returns a single element of a matrix/vector

[list_begin arguments]
[arg_def list matrix] Matrix or vector in question
[arg_def integer row] Row of the element
[arg_def integer col] Column of the element (not present for vectors)
[list_end]

[para]
[call [cmd ::math::linearalgebra::setelem] [arg matrix] [arg row] [opt col] [arg newvalue]]

Set a single element of a matrix (or vector) to a new value

[list_begin arguments]
[arg_def list matrix] [emph name] of the matrix in question
[arg_def integer row] Row of the element
[arg_def integer col] Column of the element (not present for vectors)
[list_end]

[para]
[call [cmd ::math::linearalgebra::swaprows] [arg matrix] [arg irow1] [arg irow2] [opt imin] [opt imax]]

Swap two rows in a matrix completely or only a selected part

[list_begin arguments]
[arg_def list matrix] [emph name] of the matrix in question
[arg_def integer irow1] Index of first row
[arg_def integer irow2] Index of second row
[arg_def integer imin] Minimum column index (default: 0)
[arg_def integer imin] Maximum column index (default: ncols-1)
[list_end]

[para]
[call [cmd ::math::linearalgebra::swapcols] [arg matrix] [arg icol1] [arg icol2] [opt imin] [opt imax]]

Swap two columns in a matrix completely or only a selected part

[list_begin arguments]
[arg_def list matrix] [emph name] of the matrix in question
[arg_def integer irow1] Index of first column
[arg_def integer irow2] Index of second column
[arg_def integer imin] Minimum row index (default: 0)
[arg_def integer imin] Maximum row index (default: nrows-1)
[list_end]

[list_end]



[para]
[emph "Querying matrices and vectors"]

[list_begin definitions]

[call [cmd ::math::linearalgebra::show] [arg obj] [opt format] [opt rowsep] [opt colsep]]

Return a string representing the vector or matrix, for easy printing.
(There is currently no way to print fixed sets of columns)

[list_begin arguments]
[arg_def list obj] Matrix or vector in question
[arg_def string format] Format for printing the numbers (default: %6.4f)
[arg_def string rowsep] String to use for separating rows (default: newline)
[arg_def string colsep] String to use for separating columns (default: space)
[list_end]

[para]
[call [cmd ::math::linearalgebra::dim] [arg obj]]

Returns the number of dimensions for the object (either 0 for a scalar,
1 for a vector and 2 for a matrix)

[list_begin arguments]
[arg_def any obj] Scalar, vector, or matrix
[list_end]

[para]
[call [cmd ::math::linearalgebra::shape] [arg obj]]

Returns the number of elements in each dimension for the object (either
an empty list for a scalar, a single number for a vector and a list of
the number of rows and columns for a matrix)

[list_begin arguments]
[arg_def any obj] Scalar, vector, or matrix
[list_end]

[para]
[call [cmd ::math::linearalgebra::conforming] [arg type] [arg obj1] [arg obj2]]

Checks if two objects (vector or matrix) have conforming shapes, that is
if they can be applied in an operation like addition or matrix
multiplication.

[list_begin arguments]

[arg_def string type] Type of check:
[list_begin itemized]
[item]
"shape" - the two objects have the same shape (for all element-wise
operations)
[item]
"rows" - the two objects have the same number of rows (for use as A and
b in a system of linear equations [emph "Ax = b"]
[item]
"matmul" - the first object has the same number of columns as the number
of rows of the second object. Useful for matrix-matrix or matrix-vector
multiplication.
[list_end]

[arg_def list obj1] First vector or matrix (left operand)
[arg_def list obj2] Second vector or matrix (right operand)
[list_end]

[para]
[call [cmd ::math::linearalgebra::symmetric] [arg matrix] [opt eps]]

Checks if the given (square) matrix is symmetric. The argument eps
is the tolerance.

[list_begin arguments]
[arg_def list matrix] Matrix to be inspected
[arg_def float eps] Tolerance for determining approximate equality
(defaults to 1.0e-8)
[list_end]

[list_end]



[para]
[emph "Basic operations"]

[list_begin definitions]

[call [cmd ::math::linearalgebra::norm] [arg vector] [arg type]]

Returns the norm of the given vector. The type argument can be: 1,
2, inf or max, respectively the sum of absolute values, the ordinary
Euclidean norm or the max norm.

[list_begin arguments]
[arg_def list vector] Vector, list of coefficients
[arg_def string type] Type of norm (default: 2, the Euclidean norm)
[list_end]

[call [cmd ::math::linearalgebra::norm_one] [arg vector]]

Returns the L1 norm of the given vector, the sum of absolute values

[list_begin arguments]
[arg_def list vector] Vector, list of coefficients
[list_end]

[call [cmd ::math::linearalgebra::norm_two] [arg vector]]

Returns the L2 norm of the given vector, the ordinary Euclidean norm

[list_begin arguments]
[arg_def list vector] Vector, list of coefficients
[list_end]

[call [cmd ::math::linearalgebra::norm_max] [arg vector] [opt index]]

Returns the Linf norm of the given vector, the maximum absolute
coefficient

[list_begin arguments]
[arg_def list vector] Vector, list of coefficients
[arg_def integer index] (optional) if non zero, returns a list made of the maximum
value and the index where that maximum was found.
if zero, returns the maximum value.

[list_end]

[para]
[call [cmd ::math::linearalgebra::normMatrix] [arg matrix] [arg type]]

Returns the norm of the given matrix. The type argument can be: 1,
2, inf or max, respectively the sum of absolute values, the ordinary
Euclidean norm or the max norm.

[list_begin arguments]
[arg_def list matrix] Matrix, list of row vectors
[arg_def string type] Type of norm (default: 2, the Euclidean norm)
[list_end]

[para]
[call [cmd ::math::linearalgebra::dotproduct] [arg vect1] [arg vect2]]

Determine the inproduct or dot product of two vectors. These must have
the same shape (number of dimensions)

[list_begin arguments]
[arg_def list vect1] First vector, list of coefficients
[arg_def list vect2] Second vector, list of coefficients
[list_end]

[para]
[call [cmd ::math::linearalgebra::unitLengthVector] [arg vector]]

Return a vector in the same direction with length 1.

[list_begin arguments]
[arg_def list vector] Vector to be normalized
[list_end]

[para]
[call [cmd ::math::linearalgebra::normalizeStat] [arg mv]]

Normalize the matrix or vector in a statistical sense: the mean of the
elements of the columns of the result is zero and the standard deviation
is 1.

[list_begin arguments]
[arg_def list mv] Vector or matrix to be normalized in the above sense
[list_end]

[para]
[call [cmd ::math::linearalgebra::axpy] [arg scale] [arg mv1] [arg mv2]]

Return a vector or matrix that results from a "daxpy" operation, that
is: compute a*x+y (a a scalar and x and y both vectors or matrices of
the same shape) and return the result.
[para]
Specialised variants are: axpy_vect and axpy_mat (slightly faster,
but no check on the arguments)

[list_begin arguments]
[arg_def double scale] The scale factor for the first vector/matrix (a)
[arg_def list mv1] First vector or matrix (x)
[arg_def list mv2] Second vector or matrix (y)
[list_end]

[para]
[call [cmd ::math::linearalgebra::add] [arg mv1] [arg mv2]]

Return a vector or matrix that is the sum of the two arguments (x+y)
[para]
Specialised variants are: add_vect and add_mat (slightly faster,
but no check on the arguments)

[list_begin arguments]
[arg_def list mv1] First vector or matrix (x)
[arg_def list mv2] Second vector or matrix (y)
[list_end]

[para]
[call [cmd ::math::linearalgebra::sub] [arg mv1] [arg mv2]]

Return a vector or matrix that is the difference of the two arguments
(x-y)
[para]
Specialised variants are: sub_vect and sub_mat (slightly faster,
but no check on the arguments)

[list_begin arguments]
[arg_def list mv1] First vector or matrix (x)
[arg_def list mv2] Second vector or matrix (y)
[list_end]

[para]
[call [cmd ::math::linearalgebra::scale] [arg scale] [arg mv]]

Scale a vector or matrix and return the result, that is: compute a*x.
[para]
Specialised variants are: scale_vect and scale_mat (slightly faster,
but no check on the arguments)

[list_begin arguments]
[arg_def double scale] The scale factor for the vector/matrix (a)
[arg_def list mv] Vector or matrix (x)
[list_end]

[para]
[call [cmd ::math::linearalgebra::rotate] [arg c] [arg s] [arg vect1] [arg vect2]]

Apply a planar rotation to two vectors and return the result as a list
of two vectors: c*x-s*y and s*x+c*y. In algorithms you can often easily
determine the cosine and sine of the angle, so it is more efficient to
pass that information directly.

[list_begin arguments]
[arg_def double c] The cosine of the angle
[arg_def double s] The sine of the angle
[arg_def list vect1] First vector (x)
[arg_def list vect2] Seocnd vector (x)
[list_end]

[para]
[call [cmd ::math::linearalgebra::transpose] [arg matrix]]

Transpose a matrix

[list_begin arguments]
[arg_def list matrix] Matrix to be transposed
[list_end]

[para]
[call [cmd ::math::linearalgebra::matmul] [arg mv1] [arg mv2]]

Multiply a vector/matrix with another vector/matrix. The result is
a matrix, if both x and y are matrices or both are vectors, in
which case the "outer product" is computed. If one is a vector and the
other is a matrix, then the result is a vector.

[list_begin arguments]
[arg_def list mv1] First vector/matrix (x)
[arg_def list mv2] Second vector/matrix (y)
[list_end]

[para]
[call [cmd ::math::linearalgebra::angle] [arg vect1] [arg vect2]]

Compute the angle between two vectors (in radians)

[list_begin arguments]
[arg_def list vect1] First vector
[arg_def list vect2] Second vector
[list_end]

[para]
[call [cmd ::math::linearalgebra::crossproduct] [arg vect1] [arg vect2]]

Compute the cross product of two (three-dimensional) vectors

[list_begin arguments]
[arg_def list vect1] First vector
[arg_def list vect2] Second vector
[list_end]

[para]
[call [cmd ::math::linearalgebra::matmul] [arg mv1] [arg mv2]]

Multiply a vector/matrix with another vector/matrix. The result is
a matrix, if both x and y are matrices or both are vectors, in
which case the "outer product" is computed. If one is a vector and the
other is a matrix, then the result is a vector.

[list_begin arguments]
[arg_def list mv1] First vector/matrix (x)
[arg_def list mv2] Second vector/matrix (y)
[list_end]

[list_end]



[para]
[emph "Common matrices and test matrices"]

[list_begin definitions]

[call [cmd ::math::linearalgebra::mkIdentity] [arg size]]

Create an identity matrix of dimension [emph size].

[list_begin arguments]
[arg_def integer size] Dimension of the matrix
[list_end]

[para]
[call [cmd ::math::linearalgebra::mkDiagonal] [arg diag]]

Create a diagonal matrix whose diagonal elements are the elements of the
vector [emph diag].

[list_begin arguments]
[arg_def list diag] Vector whose elements are used for the diagonal
[list_end]

[para]
[call [cmd ::math::linearalgebra::mkRandom] [arg size]]

Create a square matrix whose elements are uniformly distributed random
numbers between 0 and 1 of dimension [emph size].

[list_begin arguments]
[arg_def integer size] Dimension of the matrix
[list_end]

[para]
[call [cmd ::math::linearalgebra::mkTriangular] [arg size] [opt uplo] [opt value]]

Create a triangular matrix with non-zero elements in the upper or lower
part, depending on argument [emph uplo].

[list_begin arguments]
[arg_def integer size] Dimension of the matrix
[arg_def string uplo] Fill the upper (U) or lower part (L)
[arg_def double value] Value to fill the matrix with
[list_end]

[para]
[call [cmd ::math::linearalgebra::mkHilbert] [arg size]]

Create a Hilbert matrix of dimension [emph size].
Hilbert matrices are very ill-conditioned with respect to
eigenvalue/eigenvector problems. Therefore they
are good candidates for testing the accuracy
of algorithms and implementations.

[list_begin arguments]
[arg_def integer size] Dimension of the matrix
[list_end]

[para]
[call [cmd ::math::linearalgebra::mkDingdong] [arg size]]

Create a "dingdong" matrix of dimension [emph size].
Dingdong matrices are imprecisely represented,
but have the property of being very stable in
such algorithms as Gauss elimination.

[list_begin arguments]
[arg_def integer size] Dimension of the matrix
[list_end]

[para]
[call [cmd ::math::linearalgebra::mkOnes] [arg size]]

Create a square matrix of dimension [emph size] whose entries are all 1.

[list_begin arguments]
[arg_def integer size] Dimension of the matrix
[list_end]

[para]
[call [cmd ::math::linearalgebra::mkMoler] [arg size]]

Create a Moler matrix of size [emph size]. (Moler matrices have
a very simple Choleski decomposition. It has one small eigenvalue
and it can easily upset elimination methods for systems of linear
equations.)

[list_begin arguments]
[arg_def integer size] Dimension of the matrix
[list_end]

[para]
[call [cmd ::math::linearalgebra::mkFrank] [arg size]]

Create a Frank matrix of size [emph size]. (Frank matrices are
fairly well-behaved matrices)

[list_begin arguments]
[arg_def integer size] Dimension of the matrix
[list_end]

[para]
[call [cmd ::math::linearalgebra::mkBorder] [arg size]]

Create a bordered matrix of size [emph size]. (Bordered matrices have
a very low rank and can upset certain specialised algorithms.)

[list_begin arguments]
[arg_def integer size] Dimension of the matrix
[list_end]

[para]
[call [cmd ::math::linearalgebra::mkWilkinsonW+] [arg size]]

Create a Wilkinson W+ of size [emph size]. This kind of matrix
has pairs of eigenvalues that are very close together. Usually
the order (size) is odd.

[list_begin arguments]
[arg_def integer size] Dimension of the matrix
[list_end]

[para]
[call [cmd ::math::linearalgebra::mkWilkinsonW-] [arg size]]

Create a Wilkinson W- of size [emph size]. This kind of matrix
has pairs of eigenvalues with opposite signs, when the order (size)
is odd.

[list_begin arguments]
[arg_def integer size] Dimension of the matrix
[list_end]

[list_end]



[para]
[emph "Common algorithms"]

[list_begin definitions]

[call [cmd ::math::linearalgebra::solveGauss] [arg matrix] [arg bvect]]

Solve a system of linear equations (Ax=b) using Gauss elimination.
Returns the solution (x) as a vector or matrix of the same shape as
bvect.

[list_begin arguments]
[arg_def list matrix] Square matrix (matrix A)
[arg_def list bvect] Vector or matrix whose columns are the individual
b-vectors
[list_end]

[call [cmd ::math::linearalgebra::solvePGauss] [arg matrix] [arg bvect]]

Solve a system of linear equations (Ax=b) using Gauss elimination with
partial pivoting. Returns the solution (x) as a vector or matrix of the
same shape as bvect.

[list_begin arguments]
[arg_def list matrix] Square matrix (matrix A)
[arg_def list bvect] Vector or matrix whose columns are the individual
b-vectors
[list_end]

[para]
[call [cmd ::math::linearalgebra::solveTriangular] [arg matrix] [arg bvect] [opt uplo]]

Solve a system of linear equations (Ax=b) by backward substitution. The
matrix is supposed to be upper-triangular.

[list_begin arguments]
[arg_def list matrix] Lower or upper-triangular matrix (matrix A)
[arg_def list bvect] Vector or matrix whose columns are the individual
b-vectors
[arg_def string uplo] Indicates whether the matrix is lower-triangular
(L) or upper-triangular (U). Defaults to "U".
[list_end]

[call [cmd ::math::linearalgebra::solveGaussBand] [arg matrix] [arg bvect]]

Solve a system of linear equations (Ax=b) using Gauss elimination,
where the matrix is stored as a band matrix ([emph cf.] [sectref STORAGE]).
Returns the solution (x) as a vector or matrix of the same shape as
bvect.

[list_begin arguments]
[arg_def list matrix] Square matrix (matrix A; in band form)
[arg_def list bvect] Vector or matrix whose columns are the individual
b-vectors
[list_end]

[para]
[call [cmd ::math::linearalgebra::solveTriangularBand] [arg matrix] [arg bvect]]

Solve a system of linear equations (Ax=b) by backward substitution. The
matrix is supposed to be upper-triangular and stored in band form.

[list_begin arguments]
[arg_def list matrix] Upper-triangular matrix (matrix A)
[arg_def list bvect] Vector or matrix whose columns are the individual
b-vectors
[list_end]

[para]
[call [cmd ::math::linearalgebra::determineSVD] [arg A] [arg eps]]

Determines the Singular Value Decomposition of a matrix: A = U S Vtrans.
Returns a list with the matrix U, the vector of singular values S and
the matrix V.

[list_begin arguments]
[arg_def list A] Matrix to be decomposed
[arg_def float eps] Tolerance (defaults to 2.3e-16)
[list_end]

[para]
[call [cmd ::math::linearalgebra::eigenvectorsSVD] [arg A] [arg eps]]

Determines the eigenvectors and eigenvalues of a real
[emph symmetric] matrix, using SVD. Returns a list with the matrix of
normalized eigenvectors and their eigenvalues.

[list_begin arguments]
[arg_def list A] Matrix whose eigenvalues must be determined
[arg_def float eps] Tolerance (defaults to 2.3e-16)
[list_end]

[para]
[call [cmd ::math::linearalgebra::leastSquaresSVD] [arg A] [arg y] [arg qmin] [arg eps]]

Determines the solution to a least-sqaures problem Ax ~ y via singular
value decomposition. The result is the vector x.

[para]
Note that if you add a column of 1s to the matrix, then this column will
represent a constant like in: y = a*x1 + b*x2 + c. To force the
intercept to be zero, simply leave it out.

[list_begin arguments]
[arg_def list A] Matrix of independent variables
[arg_def list y] List of observed values
[arg_def float qmin] Minimum singular value to be considered (defaults to 0.0)
[arg_def float eps] Tolerance (defaults to 2.3e-16)
[list_end]

[para]
[call [cmd ::math::linearalgebra::choleski] [arg matrix]]

Determine the Choleski decomposition of a symmetric positive
semidefinite matrix (this condition is not checked!). The result
is the lower-triangular matrix L such that L Lt = matrix.

[list_begin arguments]
[arg_def list matrix] Matrix to be decomposed
[list_end]

[para]
[call [cmd ::math::linearalgebra::orthonormalizeColumns] [arg matrix]]

Use the modified Gram-Schmidt method to orthogonalize and normalize
the [emph columns] of the given matrix and return the result.

[list_begin arguments]
[arg_def list matrix] Matrix whose columns must be orthonormalized
[list_end]

[para]
[call [cmd ::math::linearalgebra::orthonormalizeRows] [arg matrix]]

Use the modified Gram-Schmidt method to orthogonalize and normalize
the [emph rows] of the given matrix and return the result.

[list_begin arguments]
[arg_def list matrix] Matrix whose rows must be orthonormalized
[list_end]

[para]
[call [cmd ::math::linearalgebra::dger] [arg matrix] [arg alpha] [arg x] [arg y] [opt scope]]

Perform the rank 1 operation A + alpha*x*y' inline (that is: the matrix A is adjusted).
For convenience the new matrix is also returned as the result.

[list_begin arguments]
[arg_def list matrix] Matrix whose rows must be adjusted
[arg_def double alpha] Scale factor
[arg_def list x] A column vector
[arg_def list y] A column vector
[arg_def list scope] If not provided, the operation is performed on all rows/columns of A
if provided, it is expected to be the list {imin imax jmin jmax}
where:
[list_begin itemized]
[item] [term imin] Minimum row index
[item] [term imax] Maximum row index
[item] [term jmin] Minimum column index
[item] [term jmax] Maximum column index
[list_end]
[list_end]

[para]
[call [cmd ::math::linearalgebra::dgetrf] [arg matrix]]

Computes an LU factorization of a general matrix, using partial,
pivoting with row interchanges. Returns the permutation vector.
[para]
The factorization has the form
[example {
   P * A = L * U
}]
where P is a permutation matrix, L is lower triangular with unit
diagonal elements, and U is upper triangular.
Returns the permutation vector, as a list of length n-1.
The last entry of the permutation is not stored, since it is
implicitely known, with value n (the last row is not swapped
with any other row).
At index #i of the permutation is stored the index of the row #j
which is swapped with row #i at step #i. That means that each
index of the permutation gives the permutation at each step, not the
cumulated permutation matrix, which is the product of permutations.

[list_begin arguments]
[arg_def list matrix] On entry, the matrix to be factored.
On exit, the factors L and U from the factorization
P*A = L*U; the unit diagonal elements of L are not stored.
[list_end]

[para]
[call [cmd ::math::linearalgebra::det] [arg matrix]]

Returns the determinant of the given matrix, based on PA=LU
decomposition, i.e. Gauss partial pivotal.

[list_begin arguments]
[arg_def list matrix] Square matrix (matrix A)
[arg_def list ipiv] The pivots (optionnal).
If the pivots are not provided, a PA=LU decomposition
is performed.
If the pivots are provided, we assume that it
contains the pivots and that the matrix A contains the
L and U factors, as provided by dgterf.
b-vectors
[list_end]

[para]
[call [cmd ::math::linearalgebra::largesteigen] [arg matrix] [arg tolerance] [arg maxiter]]

Returns a list made of the largest eigenvalue (in magnitude)
and associated eigenvector.
Uses iterative Power Method as provided as algorithm #7.3.3 of Golub & Van Loan.
This algorithm is used here for a dense matrix (but is usually
used for sparse matrices).

[list_begin arguments]
[arg_def list matrix] Square matrix (matrix A)
[arg_def double tolerance] The relative tolerance of the eigenvalue (default:1.e-8).
[arg_def integer maxiter] The maximum number of iterations (default:10).
[list_end]

[list_end]


[para]
[emph "Compability with the LA package"]

Two procedures are provided for compatibility with Hume's LA package:

[list_begin definitions]

[call [cmd ::math::linearalgebra::to_LA] [arg mv]]

Transforms a vector or matrix into the format used by the original LA
package.

[list_begin arguments]
[arg_def list mv] Matrix or vector
[list_end]

[call [cmd ::math::linearalgebra::from_LA] [arg mv]]

Transforms a vector or matrix from the format used by the original LA
package into the format used by the present implementation.

[list_begin arguments]
[arg_def list mv] Matrix or vector as used by the LA package
[list_end]


[list_end]

[para]


[section "STORAGE"]

While most procedures assume that the matrices are given in full form,
the procedures [emph solveGaussBand] and [emph solveTriangularBand]
assume that the matrices are stored as [emph "band matrices"]. This
common type of "sparse" matrices is related to ordinary matrices as
follows:

[list_begin itemized]
[item]
"A" is a full-size matrix with N rows and M columns.
[item]
"B" is a band matrix, with m upper and lower diagonals and n rows.
[item]
"B" can be stored in an ordinary matrix of (2m+1) columns (one for
each off-diagonal and the main diagonal) and n rows.
[item]
Element i,j (i = -m,...,m; j =1,...,n) of "B" corresponds to element
k,j of "A" where k = M+i-1 and M is at least (!) n, the number of rows
in "B".
[item]
To set element (i,j) of matrix "B" use:
[example {
    setelem B $j [expr {$N+$i-1}] $value
}]
[list_end]
(There is no convenience procedure for this yet)

[section "REMARKS ON THE IMPLEMENTATION"]

There is a difference between the original LA package by Hume and the
current implementation. Whereas the LA package uses a linear list, the
current package uses lists of lists to represent matrices. It turns out
that with this representation, the algorithms are faster and easier to
implement.

[para]
The LA package was used as a model and in fact the implementation of,
for instance, the SVD algorithm was taken from that package. The set of
procedures was expanded using ideas from the well-known BLAS library and
some algorithms were updated from the second edition of J.C. Nash's
book, Compact Numerical Methods for Computers, (Adam Hilger, 1990) that
inspired the LA package.

[para]
Two procedures are provided to make the transition between the two
implementations easier: [emph to_LA] and [emph from_LA]. They are
described above.

[section TODO]

Odds and ends: the following algorithms have not been implemented yet:
[list_begin itemized]

[item]
determineQR

[item]
certainlyPositive, diagonallyDominant
[list_end]

[section "NAMING CONFLICT"]
If you load this package in a Tk-enabled shell like wish, then the
command
[example {namespace import ::math::linearalgebra}]
results in an error
message about "scale". This is due to the fact that Tk defines all
its commands in the global namespace. The solution is to import
the linear algebra commands in a namespace that is not the global one:
[example {
package require math::linearalgebra
namespace eval compute {
    namespace import ::math::linearalgebra::*
    ... use the linear algebra version of scale ...
}
}]
To use Tk's scale command in that same namespace you can rename it:
[example {
namespace eval compute {
    rename ::scale scaleTk
    scaleTk .scale ...
}
}]

[section {BUGS, IDEAS, FEEDBACK}]

This document, and the package it describes, will undoubtedly contain
bugs and other problems.

Please report such in the category [emph {math :: linearalgebra}] of the
[uri {http://sourceforge.net/tracker/?group_id=12883} {Tcllib SF Trackers}].

Please also report any ideas for enhancements you may have for either
package and/or documentation.



[keywords math "linear algebra" vectors matrices "least squares" "linear equations"]
[manpage_end]
